Question: Lisa is $800$ meters from the base of a mountain. From where she stands, she measures the angle of elevation to the peak of the mountain to be $38^\circ$. She then walks to the base of the mountain and measures the new angle of elevation, this time getting $49^\circ$. How far is Lisa from the peak of the mountain when she is standing at its base? Do not round during your calculations. Round your final answer to the nearest meter.
Explanation: Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $AC=d$. $\angle A$ is supplementary to $49^\circ$, so $\angle A=131^\circ$. Now, because the interior angles of a triangle add to $180^\circ$, we know that $\angle C=11^\circ$. $A$ $B$ $C$ $38^\circ$ $131^\circ$ $11^\circ$ $800\text{ m}$ $d$ Since we are given one side length and all angle measures, we can use the law of sines. Using the law of sines $\begin{aligned} \dfrac{\sin(C)}{AB}&=\dfrac{\sin(B)}{AC}\\\\ \dfrac{\sin(11^\circ)}{800} &= \dfrac{\sin(38^\circ)}{d} \gray{\text{Substitute}} \\\\ d \cdot \sin(11^\circ) &= 800 \cdot \sin(38^\circ) \\\\ d &= \dfrac{800 \cdot \sin(38^\circ) }{\sin(11^\circ) } \\\\ d &\approx 2581 \,\text{m} \end{aligned}$ The answer Lisa is $2581 \,\text{m}$ from the peak of the mountain when she is standing at its base.